Journal of Lie Theory

Consider the natural action of ${\rm GL}_n({\mathbb C})$ on $p$ covectors and $q$ vectors; by Howe duality, the space of polynomial functions on this space decomposes multiplicity-free under the joint action of ${\rm GL}_n({\mathbb C})$ and $\mathfrak{gl}_{p+q}({\mathbb C})$. When $n \geq p + q$ (which is known as the stable range), the $\mathfrak{gl}_{p+q}$-modules are generalized Verma modules (GVMs, introduced by Lepowsky), on which the unipotent radical of the Hermitian real form ${\rm U}(p,q)$ of $\mathfrak{gl}_{p+q}$ acts freely. When $n p + q$, however, the structure of these modules is less transparent. Enright and Willenbring (2004) constructed resolutions for them in terms of GVMs. The goal of this paper is to exhibit a remarkable connection between these resolutions and a seemingly quite different situation, namely the $K$-type multiplicities in certain discrete series of ${\rm SU}(n, p+q)$. More precisely, we establish that the signed multiplicities of the GVMs in the resolution coincide with the values of Blattner's formula for the $K$-type multiplicities in appropriately chosen discrete series representations of ${\rm SU}(n, p+q)$.